please solve while giving clear explanations. Thanks

Consider ten independent random variables X,...,Ng which are identically distributed with an exponential distribution with expectation 4. 10 (1) Specify the approximate distribution of X = > X; . including all parameters, 1=1 using the central limit theorem. [2] (ii) Calculate the approximate value of the probability P[ X" A Suppose that x = (X1 ,X2,....*,, ) is a realisation of X1, X2..... X,. (i) (a) Derive the likelihood function L(0; >) and produce a rough sketch of its graph. (b) Use the graph produced in part (i)(a) to explain why the maximum likelihood estimate of 0 is given by () = max(x,2....,,). [4] Let Xim = max(X1.X2.....X,) be the estimator of 0, that is the random variable corresponding to *() (li] (a) Show that the cumulative distribution function of the estimator X() is given by: 2m FXml (x) = for 0 5 x50. (b) Hence, derive the probability density function of the estimator X;) - (9) Determine the expected value E(X(,)) and the variance V(X() ). (d) Show that the estimator 2n +1 2n -X(m) is an unbiased estimator of 6. [9] (iii) (@) Derive the mean square error of the estimator given in part (ii)(d). (b) Comment on the consistency of this estimator. [5] [Total 18]